The computational complexities of the above two algorithms are analyzed as follows. Specifically, the general algorithm needs to optimize more optimization variables and its complexity is given by \(\mathcal {O}\left( (N+3K)^{0.5}(N+5K)^3(J+1)^{3.5}\right)\)  based on the analytical results in [1]}, whereas the low-complexity algorithm mainly needs to solve (P5) with only one IRS phase-shift vector, which results in a complexity given by \(\mathcal {O}( (N+2.5K)^{0.5}(N+3K)^3 )\) . As such, the computational complexity is reduced by about \((J+1)^{3.5}\)  times, which is significant especially when \(J\)  is large. This is attributed to the idea drawn from Proposition REF .
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